Question

Which of the following is a vector space? For each one that isn't, show at least one criterion that it fails to meet. Assume addi- (a) The set of all position vectors (x,y,z) tion and multiplication by a scalar are that lie on the plane z=x+y defined in the obvious ways. (b) The set of all 3-vectors with vx​>0 (c) The set of all real numbers (d) The set of all possible box sizes (height, width, length)

Answer 1

A vector space is a set that meets certain criteria. The set of position vectors on a specific plane, z=x+y, is a vector space. The set of 3-vectors with a positive x-component is not a vector space.

Step-by-step explanation:

(a) The set of all position vectors (x,y,z) that lie on the plane z=x+y is a vector space. To be a vector space, it needs to satisfy certain criteria, including closure under addition and scalar multiplication. In this case, the set satisfies both criteria. For example, adding two vectors in this set will still result in a vector that lies on the plane z=x+y, and multiplying a vector by a scalar will also result in a vector on the plane.

(b) The set of all 3-vectors with vx​>0 is not a vector space because it fails to meet the criterion of closure under scalar multiplication. For example, multiplying a vector with vx​>0 by a negative scalar would result in a vector with vx​<0, which is not in the set.

(c) The set of all real numbers is not a vector space because it fails to meet the criterion of closure under addition. For example, adding two real numbers may not result in another real number.

(d) The set of all possible box sizes (height, width, length) is not a vector space as it fails to meet the criterion of closure under scalar multiplication. For example, multiplying the box sizes by a negative scalar would result in negative dimensions, which is not physically possible.